3.806 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{5/2}} \, dx\)

Optimal. Leaf size=314 \[ \frac{20 a^2 b^2 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac{2 b^4 x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{7 (a+b x)}+\frac{2 a b^3 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{a+b x}+\frac{2 b^5 B x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}-\frac{2 a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}-\frac{2 a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{\sqrt{x} (a+b x)}+\frac{10 a^3 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{a+b x} \]

[Out]

(-2*a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^(3/2)*(a + b*x)) - (2*a^4*(5*A*b +
a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(Sqrt[x]*(a + b*x)) + (10*a^3*b*(2*A*b + a*
B)*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (20*a^2*b^2*(A*b + a*B)*x^
(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (2*a*b^3*(A*b + 2*a*B)*x^(5
/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (2*b^4*(A*b + 5*a*B)*x^(7/2)*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (2*b^5*B*x^(9/2)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(9*(a + b*x))

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Rubi [A]  time = 0.344857, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{20 a^2 b^2 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac{2 b^4 x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{7 (a+b x)}+\frac{2 a b^3 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{a+b x}+\frac{2 b^5 B x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}-\frac{2 a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}-\frac{2 a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{\sqrt{x} (a+b x)}+\frac{10 a^3 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{a+b x} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^(5/2),x]

[Out]

(-2*a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^(3/2)*(a + b*x)) - (2*a^4*(5*A*b +
a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(Sqrt[x]*(a + b*x)) + (10*a^3*b*(2*A*b + a*
B)*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (20*a^2*b^2*(A*b + a*B)*x^
(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (2*a*b^3*(A*b + 2*a*B)*x^(5
/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (2*b^4*(A*b + 5*a*B)*x^(7/2)*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (2*b^5*B*x^(9/2)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(9*(a + b*x))

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Rubi in Sympy [A]  time = 34.0217, size = 301, normalized size = 0.96 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{3 a x^{\frac{3}{2}}} + \frac{512 a^{3} b \sqrt{x} \left (3 A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 \left (a + b x\right )} + \frac{256 a^{2} b \sqrt{x} \left (3 A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63} + \frac{64 a b \sqrt{x} \left (3 a + 3 b x\right ) \left (3 A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63} + \frac{160 b \sqrt{x} \left (3 A b + B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{63} + \frac{20 b \sqrt{x} \left (a + b x\right ) \left (3 A b + B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{9 a} - \frac{2 \left (3 A b + B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/x**(5/2),x)

[Out]

-A*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(3*a*x**(3/2)) + 512*a**3*b
*sqrt(x)*(3*A*b + B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(63*(a + b*x)) + 256*a**
2*b*sqrt(x)*(3*A*b + B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/63 + 64*a*b*sqrt(x)*(
3*a + 3*b*x)*(3*A*b + B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/63 + 160*b*sqrt(x)*(
3*A*b + B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/63 + 20*b*sqrt(x)*(a + b*x)*(3*
A*b + B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(9*a) - 2*(3*A*b + B*a)*(a**2 + 2
*a*b*x + b**2*x**2)**(5/2)/(a*sqrt(x))

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Mathematica [A]  time = 0.0858447, size = 122, normalized size = 0.39 \[ -\frac{2 \sqrt{(a+b x)^2} \left (21 a^5 (A+3 B x)+315 a^4 b x (A-B x)-210 a^3 b^2 x^2 (3 A+B x)-42 a^2 b^3 x^3 (5 A+3 B x)-9 a b^4 x^4 (7 A+5 B x)-b^5 x^5 (9 A+7 B x)\right )}{63 x^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^(5/2),x]

[Out]

(-2*Sqrt[(a + b*x)^2]*(315*a^4*b*x*(A - B*x) - 210*a^3*b^2*x^2*(3*A + B*x) + 21*
a^5*(A + 3*B*x) - 42*a^2*b^3*x^3*(5*A + 3*B*x) - 9*a*b^4*x^4*(7*A + 5*B*x) - b^5
*x^5*(9*A + 7*B*x)))/(63*x^(3/2)*(a + b*x))

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Maple [A]  time = 0.011, size = 140, normalized size = 0.5 \[ -{\frac{-14\,B{b}^{5}{x}^{6}-18\,A{x}^{5}{b}^{5}-90\,B{x}^{5}a{b}^{4}-126\,A{x}^{4}a{b}^{4}-252\,B{x}^{4}{a}^{2}{b}^{3}-420\,A{x}^{3}{a}^{2}{b}^{3}-420\,B{x}^{3}{a}^{3}{b}^{2}-1260\,A{x}^{2}{a}^{3}{b}^{2}-630\,B{x}^{2}{a}^{4}b+630\,Ax{a}^{4}b+126\,Bx{a}^{5}+42\,A{a}^{5}}{63\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^(5/2),x)

[Out]

-2/63*(-7*B*b^5*x^6-9*A*b^5*x^5-45*B*a*b^4*x^5-63*A*a*b^4*x^4-126*B*a^2*b^3*x^4-
210*A*a^2*b^3*x^3-210*B*a^3*b^2*x^3-630*A*a^3*b^2*x^2-315*B*a^4*b*x^2+315*A*a^4*
b*x+63*B*a^5*x+21*A*a^5)*((b*x+a)^2)^(5/2)/x^(3/2)/(b*x+a)^5

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Maxima [A]  time = 0.710085, size = 319, normalized size = 1.02 \[ \frac{2}{105} \,{\left (3 \,{\left (5 \, b^{5} x^{2} + 7 \, a b^{4} x\right )} x^{\frac{3}{2}} + 28 \,{\left (3 \, a b^{4} x^{2} + 5 \, a^{2} b^{3} x\right )} \sqrt{x} + \frac{210 \,{\left (a^{2} b^{3} x^{2} + 3 \, a^{3} b^{2} x\right )}}{\sqrt{x}} + \frac{420 \,{\left (a^{3} b^{2} x^{2} - a^{4} b x\right )}}{x^{\frac{3}{2}}} - \frac{35 \,{\left (3 \, a^{4} b x^{2} + a^{5} x\right )}}{x^{\frac{5}{2}}}\right )} A + \frac{2}{315} \,{\left (5 \,{\left (7 \, b^{5} x^{2} + 9 \, a b^{4} x\right )} x^{\frac{5}{2}} + 36 \,{\left (5 \, a b^{4} x^{2} + 7 \, a^{2} b^{3} x\right )} x^{\frac{3}{2}} + 126 \,{\left (3 \, a^{2} b^{3} x^{2} + 5 \, a^{3} b^{2} x\right )} \sqrt{x} + \frac{420 \,{\left (a^{3} b^{2} x^{2} + 3 \, a^{4} b x\right )}}{\sqrt{x}} + \frac{315 \,{\left (a^{4} b x^{2} - a^{5} x\right )}}{x^{\frac{3}{2}}}\right )} B \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/x^(5/2),x, algorithm="maxima")

[Out]

2/105*(3*(5*b^5*x^2 + 7*a*b^4*x)*x^(3/2) + 28*(3*a*b^4*x^2 + 5*a^2*b^3*x)*sqrt(x
) + 210*(a^2*b^3*x^2 + 3*a^3*b^2*x)/sqrt(x) + 420*(a^3*b^2*x^2 - a^4*b*x)/x^(3/2
) - 35*(3*a^4*b*x^2 + a^5*x)/x^(5/2))*A + 2/315*(5*(7*b^5*x^2 + 9*a*b^4*x)*x^(5/
2) + 36*(5*a*b^4*x^2 + 7*a^2*b^3*x)*x^(3/2) + 126*(3*a^2*b^3*x^2 + 5*a^3*b^2*x)*
sqrt(x) + 420*(a^3*b^2*x^2 + 3*a^4*b*x)/sqrt(x) + 315*(a^4*b*x^2 - a^5*x)/x^(3/2
))*B

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Fricas [A]  time = 0.274669, size = 161, normalized size = 0.51 \[ \frac{2 \,{\left (7 \, B b^{5} x^{6} - 21 \, A a^{5} + 9 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 63 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 210 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 315 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 63 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x\right )}}{63 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/x^(5/2),x, algorithm="fricas")

[Out]

2/63*(7*B*b^5*x^6 - 21*A*a^5 + 9*(5*B*a*b^4 + A*b^5)*x^5 + 63*(2*B*a^2*b^3 + A*a
*b^4)*x^4 + 210*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 315*(B*a^4*b + 2*A*a^3*b^2)*x^2 -
63*(B*a^5 + 5*A*a^4*b)*x)/x^(3/2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/x**(5/2),x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**(5/2), x)

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GIAC/XCAS [A]  time = 0.273403, size = 263, normalized size = 0.84 \[ \frac{2}{9} \, B b^{5} x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) + \frac{10}{7} \, B a b^{4} x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{7} \, A b^{5} x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + 4 \, B a^{2} b^{3} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + 2 \, A a b^{4} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + \frac{20}{3} \, B a^{3} b^{2} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + \frac{20}{3} \, A a^{2} b^{3} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 10 \, B a^{4} b \sqrt{x}{\rm sign}\left (b x + a\right ) + 20 \, A a^{3} b^{2} \sqrt{x}{\rm sign}\left (b x + a\right ) - \frac{2 \,{\left (3 \, B a^{5} x{\rm sign}\left (b x + a\right ) + 15 \, A a^{4} b x{\rm sign}\left (b x + a\right ) + A a^{5}{\rm sign}\left (b x + a\right )\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/x^(5/2),x, algorithm="giac")

[Out]

2/9*B*b^5*x^(9/2)*sign(b*x + a) + 10/7*B*a*b^4*x^(7/2)*sign(b*x + a) + 2/7*A*b^5
*x^(7/2)*sign(b*x + a) + 4*B*a^2*b^3*x^(5/2)*sign(b*x + a) + 2*A*a*b^4*x^(5/2)*s
ign(b*x + a) + 20/3*B*a^3*b^2*x^(3/2)*sign(b*x + a) + 20/3*A*a^2*b^3*x^(3/2)*sig
n(b*x + a) + 10*B*a^4*b*sqrt(x)*sign(b*x + a) + 20*A*a^3*b^2*sqrt(x)*sign(b*x +
a) - 2/3*(3*B*a^5*x*sign(b*x + a) + 15*A*a^4*b*x*sign(b*x + a) + A*a^5*sign(b*x
+ a))/x^(3/2)